Propositional Logic

1
Propositional Logic
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Basic Tenets

• Reasoning about the world
• Start with atomic propositions, eg.
• the sun rises in the east
• the Nile river flows north
• Law of excluded middle a proposition is either
  true or false
• Law of contradiction a proposition can never be
  true and false

3
Building complexity

• Natural language constructions
• or
• eg. the light switch is off or the light bulb
  has blown
• and
• eg. the grass is green and the sun is shining
• not
• eg. the car is not parked in the garage

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Logical Connectives

• Suppose that p and q are propositions. Construct
  formulae
• p ? q is read p or q, ? is called the
  disjunction connective
• p ? q is read p and q, ? is called the
  conjunction connective
• ?p is read not p, ? is called negation

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Other language constructions

• either or
• eg. either Harry or Cedric will win the
  Triwizard Tournament
• if then
• eg. if it starts to rain then the grass will get
  wet

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More Connectives

• Suppose that p and q are propositions
• p ? q is read p exclusive or q, ? is the
  exclusive or connective
• p ? q is read p implies q, ? is the implies
  connective

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Meaning and Truth Tables

• Denote
• TRUE ? 1
• FALSE ? 0
• What is the meaning of a logical connective?
• Recall that a proposition is either true or false
• A formula is also true or false
• The meaning of a formula is defined by the truth
  or falsity of a formula given the truth or
  falsity of the propositions it contains

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?
1
0
9
?
0
1
1
1
10
?
0
0
0
1
11
?
1
1
0
1
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Nesting

• More complex expressions are formulated by
  repeated applications of logical connectives
• eg. (p ? q) ? (?s ? q)
• Such an expression is called a propositional
  formula
• An individual proposition such as p is also a
  formula but is more often referred to as an
  atomic proposition
• A convention ranks connectives in the following
  way ?, ?, ?, ?
• eg. q ? ?s ? q is interpreted as (q ? ?s ) ? q
  and NOT q ? (?s ? q)
• Generally, if there is any chance of ambiguity,
  then use brackets!

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Propositional Formulae

• The truth of a formula is evaluated from the
  inside out
• A formula that is always true is called a
  tautology
• A formula that is always false is called a
  contradiction

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Truth tables
1
1
1
1
15
Truth tables
0
0
0
0
16
Truth tables
0
0
1
1
0
1
1
1
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The heart of a computer

• Below all the multimedia, a computer works with
  0s and 1s and logic
• A half adder adds two bits together
• c
• s

s
c
x ? y
(x ? ?y) ? (?x ? y)
y
x
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A full adder

• A full adder has a third input, a carry bit
• co ??
• s ??

s
co
x
y
ci
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Boolean Functions

• A formula like (p ? q) ? (?s ? q) can be viewed
  as a function f0, 13 ? 0, 1
• 0, 13 is short for 0, 1 ? 0, 1 ? 0, 1
• 0, 13 (u, v, w) u, v, w ? 0, 1
• there is an ordering on the arguments
• ordering made explicit by saying
  f(p, q, s) (p ? q) ? (?s ? q)
• eg. f(0, 0, 1) 0

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Boolean functions

• The logical connectives ?, ?, ? are all Boolean
  functions of two variables
• There are 16 distinct Boolean functions of two
  variables
• domain contains 4 elements, i.e., (0, 0), (0,
  1), (1, 0), (1, 1)
• each domain element can be mapped to either 0 or
  1
• there are therefore 2 x 2 x 2 x 2 24 distinct
  functions
• In general, for finite sets X and Y, the number
  of functions fX ? Y are YX

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Interesting facts

• The connectives ?, ? form a complete set
• Any Boolean function of two variables f(p, q) is
  equivalent to a formula involving p, q, ?, and ?
• ?, ? also form a complete set

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Clauses a type of formula

• A literal is either an atomic proposition or the
  negation of an atomic proposition, ie., p or ?p
• A clause is a disjunction of literals, ie.,
• where each li is a literal
• Brackets are unnecessary since there is no
  ambiguity

l1 ? l2 ? ? ln
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Clauses - examples

• p ? q
• ? p ? ? q
• p ? ? q ? r ? s
• p ? q
• ? p ? ? ? q

YES
YES
YES
NO
NO
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Conjunctive Normal Form (CNF)

• A propositional formula is in conjunctive normal
  form if it consists of a conjunction of clauses,
  ie.,
• where each ci is a clause

c1 ? c2 ? ? cn
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CNF - examples

• (p ? q)
• (? p ? ? q) ? (p ? s)
• (p ? ? q ? r ? s) ? (? p) ? (q ? r)
• (p ? q) ? s
• (? p ? ? q) ? (r ? q)

YES
YES
YES
NO
NO
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Converting a formula into CNF

• Any propositional formula can be converted to an
  equivalent formula in CNF
• 1. Eliminate connectives other than ?, ? and ?
• 2. Push ? as far as possible into the formula
• 3. Push ? as far as possible into the formula

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1. Connective Elimination

• All binary connectives are equivalent to a
  combination of ?, ?, and ?
• For example p ? q is equivalent to ?p ? q

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2. Pushing Negation

• Three laws
• ??p ? p double negation
• ?(p ? q) ? ?p ? ?q De Morgans law
• ?(p ? q) ? ?p ? ?q De Morgans law

• denotes is equivalent to or
• can be replaced by

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Pushing Negation - example
?(p ? (q ? ?r))
(?p ? ?(q ? ?r))
(?p ? (?q ? ??r))
(?p ? (?q ? r))
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3. Pushing Disjunction

• One law
• (p ? q) ? r ? (p ? r) ? (q ? r) distributive
  law
• Note that r can be a complex formula
• Like like mathematics!
• (x 3)(y 2) x(y 2) 3(y 2)
• xy 2x 3y 6

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Pushing Disjunction - example
(p ? s) ? (q ? ?r))
(p ? (q ? ?r)) ? (s ? (q ? ?r))
(p ? q) ? (p ? ?r) ? (s ? q) ? (s ? ?r)
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Reasoning

• the light switch is off or the light bulb has
  blown
• the light switch is on
• What can you conclude?

the light bulb has blown
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Reasoning

• The process of reasoning involves taking two or
  more premises and generating a conclusion
• In the previous example, the premises were of the
  form p ? q and ?p
• The conclusion was q
• This is an instance of a reasoning process called
  resolution
• Is this reasonable? Check that ((p ? q) ? ?p) ?
  q is a tautology

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Resolution Proofs

• The big question in reasoning
• If I suppose H,
• can I then conclude C?
• With resolution this question is equivalent to
• Assuming H1, H2, , and Hn and ?C
• can I derive false (the empty clause)?
• H1 ? H2 ? ? Hn is the CNF equivalent of H
• This is not trivial!

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Resolution

• A resolution step takes two clauses where an atom
  is positive in one clause and negative in the
  other
• The derived conclusion is the combination of two
  clauses without the specified atom

p1 ? p2 ? ? pn ? p
?p ? q1 ? q2 ? ? qm
p1 ? p2 ? ? pn ? q1 ? q2 ? ? qm
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Resolution Proof

• The premises for a resolution step comes from two
  sources
• 1. Any assumption Hi or ?C
• 2. The conclusion from other resolution steps
• Repeatedly use resolution until you can derive
  the empty clause then conclude C
• The empty clause is also called false because
• p ? p ? FALSE

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Resolution Proofs - example

• the light switch is off or the light bulb has
  blown
• the light switch is on
• Conclude
• the light bulb has blown

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Resolution Proofs - example
the light switch is off ? the light bulb has
blown
the light switch is on
?the light bulb has blown
the light bulb has blown

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Putting it all together

• Prove by resolution that
• s ? (?s ? (r ? (p ? ?r))) ? (?r ? q)
• implies
• p ? q

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Putting it all together

• Firstly, convert to CNF
• Let H s ? (?s ? (r ? (p ? ?r))) ? (?r ? q)
• s ? (?s ? r) ? (?s ? p ? ?r)
• ? (?r ? q)
• Negate the conclusion and convert to CNF
• Let C p ? q. Therefore ?C ?(p ? q)
• (?p ? ?q)

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The Proof
H s ? (?s ? r) ? (?s ? p ? ?r) ? (?r ?
q)
r
?C (?p ? ?q)
q
?p
?s ? ?r
?r

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The End

• Therefore
• s ? (?s ? (r ? (p ? ?r))) ? (?r ? q)
• implies
• p ? q